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If the tangent at the point P(2,4) to th...

If the tangent at the point `P(2,4)` to the parabola `y^2=8x` meets the parabola `y^2=8x+5` at `Qa n dR ,` then find the midpoint of chord `Q Rdot`

A

`(2, 4)`

B

`(4, 2)`

C

`(7, 9)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
The equation of the tangent to `y^(2)=8x" at "P(2, 4)` is
`4y=4(x+2)rArrx-y+2=0" …(i)"`
Let `(x_(1), y_(1))` be the mid-point of chord QR. Then, equation of QR is
`yy_(1)-4(x+x_(1))-5=y_(1)^(2)-x_(1)-5`
`rArr" "4x-yy_(1)-4x_(1)+y_(1)^(2)=0" ...(ii)"`
Clearly, (i) and (ii) represent the same line.
.`:." "4/1=-y_(1)/(-1)=(-4x_(1)+y_(1)^(2))/2`
`rArr" "y_(1)=4" and "8=-4x_(1)+y_(1)^(2)rArry_(1)=4" and "x_(1)=2`
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OBJECTIVE RD SHARMA ENGLISH-PARABOLA-Chapter Test
  1. If the tangent at the point P(2,4) to the parabola y^2=8x meets the pa...

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  2. If y=2x+k is a tangent to the curve x^(2)=4y, then k is equal to

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  3. The normal drawn at a point (a t1 2,-2a t1) of the parabola y^2=4a x m...

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  4. The mid-point of the chord 2x+y-4=0 of the parabola y^(2)=4x is

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  5. The two ends of latusrectum of a parabola are the points (3, 6) and (-...

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  6. Prove that the locus of the middle points of all chords of the parabol...

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  7. The focus of the parabola x^2-8x+2y+7=0 is

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  8. The point of contact of the line x-2y-1=0 with the parabola y^(2)=2(x-...

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  9. Find the number of distinct normals that can be drawn from (-2,1) to t...

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  10. At what point on the parabola y^2=4x the normal makes equal angle with...

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  11. Three normals to the parabola y^2= x are drawn through a point (C, O) ...

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  12. The normal chord of a parabola y^2= 4ax at the point P(x1, x1) subten...

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  13. AB, AC are tangents to a parabola y^2=4ax; p1, p2, p3 are the lengths...

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  14. The circles on the focal radii of a parabola as diameter touch: A) th...

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  15. If the normals from any point to the parabola y^2=4x cut the line x=2 ...

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  16. about to only mathematics

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  17. The equation of the tangent to the parabola y^(2)=8x which is perpendi...

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  18. the tangent drawn at any point P to the parabola y^2= 4ax meets the di...

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  19. about to only mathematics

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  20. The parabola y^(2)=4ax passes through the point (2,-6). Find the lengt...

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  21. A variable circle passes through the fixed point (2, 0) and touches y-...

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